Perturbing eigenvalues of nonnegative centrosymmetric matrices

TL;DR

This paper investigates the spectral properties of nonnegative centrosymmetric matrices, demonstrating how to modify a few eigenvalues without affecting the rest or the matrix's structure, and provides algorithms for these modifications.

Contribution

It introduces methods to alter specific eigenvalues of nonnegative centrosymmetric matrices while preserving their structure and nonnegativity, partially answering existing open questions.

Findings

Ability to change one, two, or three eigenvalues without affecting others

Preservation of nonnegativity and centrosymmetry during eigenvalue modification

Development of algorithmic procedures for eigenvalue adjustments

Abstract

An $n\times n$ matrix $C$ is said to be {\it centrosymmetric} if it satisfies the relation $JCJ=C$, where $J$ is the $n\times n$ counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively in the literature. Many results for centrosymmetric matrices have been generalized to wider classes of matrices that arise in a wide variety of disciplines. In this paper, we obtain interesting spectral properties for nonnegative centrosymmetric matrices. We show how to change one single eigenvalue, two or three eigenvalues of an $n\times n$ nonnegative centrosymmetric matrix without changing any of the remaining eigenvalues neither nonnegativity nor the centrosymmetric structure. Moreover, our results allow partially answer some known questions given by Guo [11] and by Guo and Guo [12]. Our proofs generate algorithmic procedures that allow to compute a solution matrix.